Theorem breq12 4009

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4007	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4008	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   class class class wbr 4004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005

This theorem is referenced by:  breq12i  4013  breq12d  4017  breqan12d  4020  posng  4699  isopolem  5823  poxp  6233  rbropapd  6243  ecopover  6633  ecopoverg  6636  ltdcnq  7396  recexpr  7637  ltresr  7838  reapval  8533  ltxr  9775  xrltnr  9779  xrltnsym  9793  xrlttr  9795  xrltso  9796  xrlttri3  9797  xposdif  9882  f1olecpbl  12734  exmidsbthrlem  14773